Encyclopedia of Microtonal Music Theory

TM-Reduced Lattice Basis

"T" stands for Tenney, "M" for Minkowski. A method for reducing the basis of a lattice. First we need to define Tenney height. if p / q is a positive rational number in reduced form, then the Tenney height is TH(p / q) = p · q.
Now suppose {q1, ..., qn} are n multiplicatively linearly independent positive rational numbers. Linear independence can be equated, for instance, with the condition that rank of the matrix whose rows are the monzos for qi is n. Then {q1, ..., qn} is a basis for a lattice
L, consisting of every positive rational number of the form q1e1 ... q1en where the ei are integers and where the log of the Tenney height defines a norm. Let t1 > 1 be the shortest (in terms of Tenney height) rational number in L greater than 1. Define ti > 1 inductively
as the shortest number in L independent of {t1, ... ti-1} and such that {t1, ..., ti} can be extended to be a basis for L. In this way we obtain {t1, ..., tn}, the TM reduced basis of L. See this definition of Minkowski reduction and definitions by Gene Ward Smith.

5-Limit Base Examples

Here are some of the 5-limit lattice bases and the resulting periodicity blocks, rendered in Tonalsoft™ Tonescape™. The thickest pink lines are the two unison-vectors which form the lattice basis, the cubes represent the exponents of the prime-factors 3 and 5, which designate the ratios in the just intonation version of the tuning, and the thin pink lines connect the ratios together, showing the periodicity-block structure. Note that in the rectangular lattices the pink connectors represent only the 3 and 5 axes, while in the triangular lattices there is also a third connector which represents an axis containing both 3 and 5. Note also that the toroidal lattices cannot show the unison-vectors, because in that geometry the unison-vectors are reduced to a point.

12 ET/EDO

Enharmonic Diesis

27 30 5-3

[ 7, 0, -3>

128 / 125

41.05885841

Syntonic Comma

2-4 34 5-1

[-4 4, -1>

81 / 80

21.5062896

rectangular

triangular

toroidal

15 ET/EDO

Maximal Diesis

21 3-5 53

[ 1 -5, 3>

250 / 243

49.16613727

Enharmonic Diesis

27 30 5-3

[ 7, 0, -3>

128 / 125

41.05885841

rectangular

triangular

toroidal

19 ET/EDO

Magic Comma

2-10 3-1 55

[-10 -1, 5>

3125 / 3072

29.61356846

Syntonic Comma

2-4 34 5-1

[-4 4, -1>

81 / 80

21.5062896

rectangular

triangular

toroidal

31 ET/EDO

Syntonic Comma

2-4 34 5-1

[-4 4, -1>

81 / 80

21.5062896

217 31 5-8

[17 1, -8>

393,216 / 390,625

11.44528995

rectangular

triangular

toroidal

34 ET/EDO

Diaschisma

211 3-4 5-2

[11 -4, -2>

2048 / 2025

19.55256881

Kleisma

2-6 3-5 56

[-6 -5, 6>

15,625 / 15,552

8.107278862

rectangular

triangular

toroidal

53 ET/EDO

Kleisma

2-6 3-5 56

[-6 -5, 6>

15,625 / 15,552

8.107278862

Schisma

2-15 38 51

[-15, 8, 1>

32,805 / 32,768

1.953720788

rectangular

triangular

toroidal

65 ET/EDO

22 39 5-7

[ 2, 9, -7>

78,732 / 78,125

13.39901073

Schisma

2-15 38 51

[-15, 8, 1>

32,805 / 32,768

1.953720788

118 ET/EDO

28 314 5-13

[ 8, 14, -13>

1,792,620 / 1,787,149

5.291731873

Schisma

2-15 38 51

[-15, 8, 1>

32,805 / 32,768

1.953720788

rectangular

triangular

toroidal

171 ET/EDO

Schisma

2-15 38 51

[-15, 8, 1>

32,805 / 32,768

1.953720788

21 3-27 518

[ 1, -27, 18>

4,711,802 / 4,711,802

0.861826202

rectangular

triangular

toroidal

289 ET/EDO

27 341 5-31

[ 7, 41, -31>

5,146,069 / 5,132,918

4.429905671

Schisma

2-15 38 51

[-15, 8, 1>

32,805 / 32,768

1.953720788

441 ET/EDO

238 3-2 5-15

[38, -2, -15>

6,719,816 / 6,714,445

1.384290297

21 3-27 518

[ 1, -27, 18>

4,711,802 / 4,711,802

0.861826202

559 ET/EDO

238 3-2 5-15

[38, -2, -15>

6,719,816 / 6,714,445

1.384290297

2-16 335 5-17

[-16, 35, -17>

6,437,705 / 6,433,646

1.091894586

612 ET/EDO

21 3-27 518

[ 1, -27, 18>

4,711,802 / 4,711,802

0.861826202

2-53 310 516

[-53, 10, 16>

4,758,837 / 4,757,272

0.569430491

toroidal top view

toroidal close up

toroidal side-view close up

When the cardinality of the EDO gets this high, it is difficult to see a difference in the geometry of their toruses, so graphics for the following are omitted.

730 ET/EDO

2-16 335 5-17

[-16, 35, -17>

6,437,705 / 6,433,646

1.091894586

2-53 310 516

[-53, 10, 16>

4,758,837 / 4,757,272

0.569430491

1,171 ET/EDO and Above

additional 5-limit lattice bases

1,171 ET / EDO

237 325 5-33

254 32 5-37

1,783 ET / EDO

254 32 5-37

2-90 3-15 549

2,513 ET / EDO

2-107 347 514

2-17 362 5-35

4,296 ET / EDO

271 337 5-99

2-90 3-15 549

6,809 ET / EDO

2-178 3146 523

2-90 3-15 549

16,572 ET / EDO

292 3191 5-170

2161 3-81 5-12

20,868 ET / EDO

2161 3-81 5-12

221 3290 5-207

25,164 ET / EDO

2-111 3-305 5256

2161 3-81 5-12

52,841 ET / EDO

221 3290 5-207

2-412 3153 573

73,709 ET / EDO

221 3290 5-207

2-573 3237 585

78,005 ET / EDO

2140 3195 5-374

2-573 3237 585

. . . . . . . . .

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